# Inequalities for the differential subordinates of Martingales, harmonic functions and Ito processes

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 Title: Inequalities for the differential subordinates of Martingales, harmonic functions and Ito processes Author(s): Choi, Changsun Doctoral Committee Chair(s): Ruan, Zhong-Jin Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: In Chapter 1 we sharpen Burkholder's inequality $\mu(\vert v\vert\geq1)\leq2\Vert u\Vert\sb1$ for two harmonic functions u and v by adjoining an extra assumption. That is, we prove the weak-type inequality $\mu(\vert v\vert\geq1)\leq K\Vert u\Vert\sb1$ under the assumptions that $\vert v(\xi)\vert\leq\vert u(\xi)\vert, \vert\nabla v\vert\leq\vert\nabla u\vert$ and the extra assumption that $\nabla u\cdot\nabla v$ = 0. Here $\mu$ is the harmonic measure with respect to $\xi$ and the constant 1 \$